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Exponential integrators

  • Marlis Hochbruck (a1) and Alexander Ostermann (a2)
  • DOI: http://dx.doi.org/10.1017/S0962492910000048
  • Published online: 01 May 2010
Abstract

In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.

Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

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L. Bergamaschi and M. Vianello (2000), ‘Efficient computation of the exponential operator for large, sparse, symmetric matrices’, Numer. Linear Algebra Appl. 7, 2745.

H. Berland , A. L. Islas and C. M. Schober (2007 a), ‘Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation’, J. Comput. Phys. 225, 284299.

H. Berland , B. Owren and B. Skaflestad (2006), ‘Solving the nonlinear Schrödinger equation using exponential integrators’, Model. Identif. Control 27, 201217.

H. Berland , B. Skaflestad and W. M. Wright (2007 b), ‘EXPINT: A MATLAB package for exponential integrators’, ACM Trans. Math. Software 33, 4: 1–4: 17.

G. Beylkin , J. M. Keiser and L. Vozovoi (1998), ‘A new class of time discretization schemes for the solution of nonlinear PDEs’, J. Comput. Phys. 147, 362387.

J. J. Biesiadecki and R. D. Skeel (1993), ‘Dangers of multiple time step methods’, J. Comput. Phys. 109, 318328.

S. Blanes , F. Casas and J. Ros (2002), ‘High order optimized geometric integrators for linear differential equations’, BIT 42, 262284.

S. Blanes , F. Casas , J. Oteo and J. Ros (2009), ‘The Magnus expansion and some of its applications’, Physics Reports 470, 151238.

M. A. Botchev , I. Faragó and R. Horváth (2009), ‘Application of operator splitting to the Maxwell equations including a source term’, Appl. Numer. Math. 59, 522541.

M. A. Botchev , D. Harutyunyan and J. J. W. van der Vegt (2006), ‘The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations’, J. Comput. Phys. 216, 654686.

M. Caliari (2007), ‘Accurate evaluation of divided differences for polynomial interpolation of exponential propagators’, Computing 80, 189201.

M. Caliari and A. Ostermann (2009), ‘Implementation of exponential Rosenbrocktype integrators’, Appl. Numer. Math. 59, 568581.

M. Caliari , M. Vianello and L. Bergamaschi (2004), ‘Interpolating discrete advection-diffusion propagators at Leja sequences’, J. Comput. Appl. Math. 172, 7999.

M. P. Calvo and C. Palencia (2006), ‘A class of explicit multistep exponential integrators for semilinear problems’, Numer. Math. 102, 367381.

E. Celledoni , D. Cohen and B. Owren (2008), ‘Symmetric exponential integrators with an application to the cubic Schrödinger equation’, Found. Comp. Math. 8, 303317.

E. Celledoni , A. Marthinsen and B. Owren (2003), ‘Commutator-free Lie group methods’, Future Generation Computer Systems 19, 341352.

D. Cohen , T. Jahnke , K. Lorenz and C. Lubich (2006), Numerical integrators for highly oscillatory Hamiltonian systems: A review. In Analysis, Modeling and Simulation of Multiscale Problems (A. Mielke , ed.), Springer, pp. 553576.

M. Condon , A. Deaño and A. Iserles (2009), ‘On highly oscillatory problems arising in electronic engineering’, Mathematical Modelling and Numerical Analysis 43, 785804.

S. M. Cox and P. C. Matthews (2002), ‘Exponential time differencing for stiff systems’, J. Comput. Phys. 176, 430455.

P. Deuflhard (1979), ‘A study of extrapolation methods based on multistep schemes without parasitic solutions’, Z. Angew. Math. Phys. 30, 177189.

P. Deuflhard , J. Hermans , B. Leimkuhler , A. Mark , S. Reich and R. D. Skeel , eds (1999), Algorithms for Macromolecular Modelling, Vol. 4 of Lecture Notes in Computational Science and Engineering, Springer.

J. Dixon and S. McKee (1986), ‘Weakly singular discrete Gronwall inequalities’, Z. Angew. Math. Mech. 66, 535544.

V. L. Druskin and L. A. Knizhnerman (1994), ‘On application of the Lanczos method to solution of some partial differential equations’, J. Comput. Appl. Math. 50, 255’262.

V. L. Druskin and L. A. Knizhnerman (1995), ‘Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic’, Numer. Linear Algebra Appl. 2, 205217.

B. L. Ehle and J. D. Lawson (1975), ‘Generalized Runge-Kutta processes for stiff initial-value problems’, J. Inst. Math. Appl. 16, 1121.

M. Eiermann and O. G. Ernst (2006), ‘A restarted Krylov subspace method for the evaluation of matrix functions’, SIAM J. Numer. Anal. 44, 24812504.

E. Emmrich (2005), ‘Stability and error of the variable two-step BDF for semilinear parabolic problems’, J. Appl. Math. Comput. 19, 3355.

A. Friedli (1978), Verallgemeinerte Runge-Kutta Verfahren zur Lösung steifer Differentialgleichungssysteme. In Numerical Treatment of Differential Equations (R. Burlirsch , R. Grigorieff and J. Schröder , eds), Vol. 631 of Lecture Notes in Mathematics, Springer, pp. 3550.

R. A. Friesner , L. S. Tuckerman , B. C. Dornblaser and T. V. Russo (1989), ‘A method for exponential propagation of large systems of stiff nonlinear differential equations’, J. Sci. Comput. 4, 327354.

E. Gallopoulos and Y. Saad (1992), ‘Efficient solution of parabolic equations by Krylov approximation methods’, SIAM J. Sci. Statist. Comput. 13, 12361264.

B. García-Archilla , J. M. Sanz-Serna and R. D. Skeel (1998), ‘Long-time-step methods for oscillatory differential equations’, SIAM J. Sci. Comput. 20, 930963.

W. Gautschi (1961), ‘Numerical integration of ordinary differential equations based on trigonometric polynomials’, Numer. Math. 3, 381397.

M. A. Gondal (2010), ‘Exponential Rosenbrock integrators for option pricing’, J. Comput. Appl. Math. 234, 11531160.

C. González and M. Thalhammer (2007), ‘A second-order Magnus-type integrator for quasi-linear parabolic problems’, Math. Comp. 76, 205231.

C. González , A. Ostermann and M. Thalhammer (2006), ‘A second-order Magnustype integrator for nonautonomous parabolic problems’, J. Comput. Appl. Math. 189, 142156.

V. Grimm (2005 a), ‘A note on the Gautschi-type method for oscillatory secondorder differential equations’, Numer. Math. 102, 6166.

V. Grimm (2005 b), ‘On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations’, Numer. Math. 100, 7189.

V. Grimm and M. Hochbruck (2006), ‘Error analysis of exponential integrators for oscillatory second-order differential equations’, J. Phys. A 39, 54955507.

V. Grimm and M. Hochbruck (2008), ‘Rational approximation to trigonometric operators’, BIT 48, 215229.

H. Grubmüller , H. Heller , A. Windemuth and K. Schulten (1991), ‘Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions’, Molecular Simulation 6, 121142.

E. Hairer and C. Lubich (2000), ‘Long-time energy conservation of numerical methods for oscillatory differential equations’, SIAM J. Numer. Anal. 38, 414441.

J. Hersch (1958), ‘Contribution à la méthode des équations aux différences’, Z. Angew. Math. Phys. 9, 129180.

N. J. Higham (2008), Functions of Matrices: Theory and Computation, SIAM.

N. J. Higham and A. H. Al-Mohy (2010), Computing matrix functions. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 159208.

M. Hochbruck and C. Lubich (1997), ‘;On Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. 34, 19111925.

M. Hochbruck and C. Lubich (1999 b), ‘Exponential integrators for quantumclassical molecular dynamics’, BIT 39, 620645.

M. Hochbruck and C. Lubich (1999 c), ‘A Gautschi-type method for oscillatory second-order differential equations’, Numer. Math. 83, 403426.

M. Hochbruck and C. Lubich (2003), ‘On Magnus integrators for time-dependent Schrödinger equations’, SIAM J. Numer. Anal. 41, 945963.

M. Hochbruck and A. Ostermann (2005 a), ‘Explicit exponential Runge-Kutta methods for semilinear parabolic problems’, SIAM J. Numer. Anal. 43, 10691090.

M. Hochbruck and A. Ostermann (2005 b), ‘Exponential Runge-Kutta methods for parabolic problems’, Appl. Numer. Math. 53, 323339.

M. Hochbruck , M. Hönig and A. Ostermann (2009 a), ‘A convergence analysis of the exponential Euler iteration for nonlinear ill-posed problems’, Inverse Problems 25, 075009.

M. Hochbruck , M. Hönig and A. Ostermann (2009 b), ‘Regularization of nonlinear ill-posed problems by exponential integrators’, Mathematical Modelling and Numerical Analysis 43, 709720.

M. Hochbruck , C. Lubich and H. Selhofer (1998), ‘Exponential integrators for large systems of differential equations’, SIAM J. Sci. Comput. 19, 15521574.

M. Hochbruck , A. Ostermann and J. Schweitzer (2009 c), ‘Exponential Rosenbrock-type methods’, SIAM J. Numer. Anal. 47, 786803.

A. Iserles (2002 a), ‘On the global error of discretization methods for highlyoscillatory ordinary differential equations’, BIT 42, 561599.

A. Iserles (2002 b), ‘Think globally, act locally: Solving highly-oscillatory ordinary differential equations’, Appl. Numer. Math. 43, 145160.

A. Iserles and S. Nørsett (2004), ‘On quadrature methods for highly oscillatory integrals and their implementation’, BIT 44, 755772.

A. Iserles , H. Z. Munthe-Kaas , S. P. Nørsett and A. Zanna (2000), Lie-group methods. In Acta Numerica, Vol. 9, Cambridge University Press, pp. 215365.

J. A. Izaguirre , S. Reich and R. D. Skeel (1999), ‘Longer time steps for molecular dynamics’, J. Chem. Phys. 110, 98539864.

T. Jahnke (2004), ‘Long-time-step integrators for almost-adiabatic quantum dynamics’, SIAM J. Sci. Comput. 25, 21452164.

T. Jahnke and C. Lubich (2000), ‘Error bounds for exponential operator splittings’, BIT 40, 735744.

T. Jahnke and C. Lubich (2003), ‘Numerical integrators for quantum dynamics close to the adiabatic limit’, Numer. Math. 94, 289314.

C. Karle , J. Schweitzer , M. Hochbruck and K.-H. Spatschek (2008), ‘A parallel implementation of a two-dimensional fluid laser-plasma integrator for stratified plasma-vacuum systems’, J. Comput. Phys. 227, 77017719.

C. Karle , J. Schweitzer , M. Hochbruck , E. W. Laedke and K.-H. Spatschek (2006), ‘Numerical solution of nonlinear wave equations in stratified dispersive media’, J. Comput. Phys. 216, 138152.

A.-K. Kassam and L. N. Trefethen (2005), ‘Fourth-order time-stepping for stiff PDEs’, SIAM J. Sci. Comput. 26, 12141233.

L. Knizhnerman and V. Simoncini (2009), ‘A new investigation of the extended Krylov subspace method for matrix function evaluations’, Numer. Linear Algebra Appl. In press.

R. Kosloff (1994), ‘Propagation methods for quantum molecular dynamics’, Annu. Rev. Phys. Chem. 45, 145178.

S. Krogstad (2005), ‘Generalized integrating factor methods for stiff PDEs’, J. Comput. Phys. 203, 7288.

J. D. Lambert and S. T. Sigurdsson (1972), ‘Multistep methods with variable matrix coefficients’, SIAM J. Numer. Anal. 9, 715733.

J. D. Lawson (1967), ‘Generalized Runge-Kutta processes for stable systems with large Lipschitz constants’, SIAM J. Numer. Anal. 4, 372380.

M. López-Fernández , C. Palencia and A. Schädle (2006), ‘A spectral order method for inverting sectorial Laplace transforms’, SIAM J. Numer. Anal. 44, 13321350.

K. Lorenz , T. Jahnke and C. Lubich (2005), ‘Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition’, BIT 45, 91115.

Q. Ma and J. A. Izaguirre (2003 a), Long time step molecular dynamics using targeted Langevin stabilization. In SAC '03: Proc. 2003 ACM Symposium on Applied Computing, ACM, New York, pp. 178182.

Q. Ma and J. A. Izaguirre (2003 b), ‘Targeted mollified impulse: A multiscale stochastic integrator for long molecular dynamics simulations’, Multiscale Model. Simul. 2, 121.

Q. Ma , J. A. Izaguirre and R. D. Skeel (2003), ‘Verlet-I/R-RESPA/impulse is limited by nonlinear instabilities’, SIAM J. Sci. Comput. 24, 19511973.

W. Magnus (1954), ‘On the exponential solution of differential equations for a linear operator’, Comm. Pure Appl. Math. 7, 649673.

A. Martínez , L. Bergamaschi , M. Caliari and M. Vianello (2009), ‘A massively parallel exponential integrator for advection-diffusion models’, J. Comput. Appl. Math. 231, 8291.

R. I. McLachlan and G. R. W. Quispel (2002), Splitting methods. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 341434.

P. C. Moan and J. Niesen (2008), ‘Convergence of the Magnus series’, Found. Comput. Math. 8, 291301.

C. Moler and C. Van Loan (2003), ‘Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later’, SIAM Rev. 45, 349.

I. Moret and P. Novati (2001), ‘An interpolatory approximation of the matrix exponential based on Faber polynomials’, J. Comput. Appl. Math. 131, 361380.

I. Moret and P. Novati (2004), ‘RD-rational approximations of the matrix exponential’, BIT 44, 595615.

A. Nauts and R. E. Wyatt (1983), ‘New approach to many-state quantum dynamics: The recursive-residue-generation method’, Phys. Rev. Lett. 51, 22382241.

P. Nettesheim , F. A. Bornemann , B. Schmidt and C. Schütte (1996), ‘An explicit and symplectic integrator for quantum-classical molecular dynamics’, Chem. Phys. Lett. 256, 581588.

S. P. Nørsett (1969), An A-stable modification of the Adams-Bashforth methods. In Conference on the Numerical Solution of Differential Equations, Vol. 109 of Lecture Notes in Mathematics, Springer, pp. 214219.

A. Ostermann , M. Thalhammer and W. M. Wright (2006), ‘A class of explicit exponential general linear methods’, BIT 46, 409431.

T. J. Park and J. C. Light (1986), ‘Unitary quantum time evolution by iterative Lanczos reduction’, J. Chem. Phys. 85, 58705876.

U. Peskin , R. Kosloff and N. Moiseyev (1994), ‘The solution of the time dependent Schrödinger equation by the (t, t′) method: The use of global polynomial propagators for time dependent Hamiltonians’, J. Chem. Phys. 100, 88498855.

M. Pototschnig , J. Niegemann , L. Tkeshelashvili and K. Busch (2009), ‘Timedomain simulations of nonlinear Maxwell equations using operator-exponential methods’, IEEE Trans. Antenn. Propag. 57, 475483.

N. Rambeerich , D. Y. Tangman , A. Gopaul and M. Bhuruth (2009), ‘Exponential time integration for fast finite element solutions of some financial engineering problems’, J. Comput. Appl. Math. 224, 668678.

Y. Saad (1992), ‘Analysis of some Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. 29, 209228.

Y. Saad (2003), Iterative Methods for Sparse Linear Systems, 2nd edn, SIAM.

A. Schädle , M. López-Fernández and C. Lubich (2006), ‘Fast and oblivious convolution quadrature’, SIAM J. Sci. Comput. 28, 421438.

T. Schlick , R. D. Skeel , A. T. Brunger , L. V. Kalé , J. A. Board , J. Hermans and K. Schulten (1999), ‘Algorithmic challenges in computational molecular biophysics’, J. Comput. Phys. 151, 948.

C. Schütte and F. A. Bornemann (1999), ‘On the singular limit of the quantumclassical molecular dynamics model’, SIAM J. Appl. Math. 59, 12081224.

R. B. Sidje (1998), ‘Expokit: A software package for computing matrix exponentials’, ACM Trans. Math. Software 24, 130156.

D.E. Stewart and T. S. Leyk (1996), ‘Error estimates for Krylov subspace approximations of matrix exponentials’, J. Comput. Appl. Math. 72, 359369.

K. Strehmel and R. Weiner (1987), ‘B-convergence results for linearly implicit one step methods’, BIT 27, 264281.

K. Strehmel and R. Weiner (1992), Linear-implizite Runge-Kutta Methoden und ihre Anwendungen, Vol. 127 of Teubner-Texte zur Mathematik, Teubner.

H. Tal-Ezer and R. Kosloff (1984), ‘An accurate and efficient scheme for propagating the time-dependent Schrödinger equation’, J. Chem. Phys. 81, 39673971.

H. Tal-Ezer , R. Kosloff and C. Cerjan (1992), ‘Low-order polynomial approximation of propagators for the time-dependent Schrödinger equation’, J. Comput. Phys. 100, 179187.

D. Y. Tangman , A. Gopaul and M. Bhuruth (2008), ‘Exponential time integration and Chebychev discretisation schemes for fast pricing of options’, Appl. Numer. Math. 58, 13091319.

U. Tautenhahn (1994), ‘On the asymptotical regularization of nonlinear ill-posed problems’, Inverse Problems 10, 14051418.

M. Thalhammer (2006), ‘A fourth-order commutator-free exponential integrator for nonautonomous differential equations’, SIAM J. Numer. Anal. 44, 851864.

M. Tokman (2006), ‘Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods’, J. Comput. Phys. 213, 748776.

M. Tokman and P. M. Bellan (2002), ‘Three-dimensional model of the structure and evolution of coronal mass ejections’, Astrophys. J. 567, 12021210.

L. N. Trefethen , J. A. C. Weideman and T. Schmelzer (2006), ‘Talbot quadratures and rational approximations’, BIT 46, 653670.

M. Tuckerman , B. J. Berne and G. J. Martyna (1992), ‘Reversible multiple time scale molecular dynamics’, J. Chem. Phys. 97, 19902001.

J. Verwer (1976), ‘On generalized linear multistep methods with zero-parasitic roots and an adaptive principal root’, Numer. Math. 27, 143155.

J. G. Verwer and M. A. Botchev (2009), ‘Unconditionally stable integration of Maxwell's equations’, Linear Algebra Appl. 431, 300317.

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